We are updating our kitchen and I was just wondering if someone could tell me the code appropriate type of circuit breakers to use for circuits in the kitchen.
Looks like NY State is 2017 NEC, City is 2008, so State you will need to use AFCI breakers, city normal 20A breakers will be acceptable. A city outside of NYC may also have revisions.
Small Appliance Branch Circuits (that serve countertop surfaces) are required to be 20A. The Code book says the two or more SABC's "shall serve all wall and floor" receptacles in kitchen, dining, pantry, and similar areas and not serve other areas. Some on this site will argue dining receptacles can be 15A circuits, I don't see how they get that.
NEC 2017 has is an exception that allows additional 15A circuits for outlets that are for dedicated appliances, the 2008 exception for applies to refrigeration outlets only.
Sharing appliances on circuits gets confusing and hard to generalize. For instance a fixed in place limitation of 50% would allow two receptacles to feed a two 9A fixed in place appliances to share a 20A circuit, but an 11A and 4A can't share a receptacle circuit. Less amps, but you can't do it. So to generalize is hard.
Also you asked about breakers, not gfci, but 2008 only kitchen countertop receptacles required gfci protection, the 2017 edition also requires within 6' of a sink, no fridge exception.
First the breaker must match your panel (Square D in a Square D panel; Eaton BR in a Challenger panel; sometimes it's tricky).
Then, the breaker must be the correct ampacity for the wire. So if the wire is #14 use a 15A breaker. Do not use a 20A breaker unless all wire in the circuit is known to be #12 copper or #10 aluminum.
If it is feasible for you to add new circuits to the kitchen, then there are a minimum number of circuits which must exist, and a few more we recommend for usability's sake. The Code requirements are slumlord minimums; exceeding the minimum makes a happier kitchen.
You must install at least two 20A receptacle circuits, which serve receptacles only, in kitchen countertops, rest of kitchen, pantry, dining area, breakfast nook etc. These circuits can also serve small loads on a gas range (i.e. spark ignitors) and also a wall clock. The exhaust fan CANNOT be on this circuit.
Electric ranges and ovens can share a dedicated circuit (or not), and are subject to special/unusual math as to circuit ampacity. Generally a 50A circuit/breaker is a safe choice, but the best choice is conduit between range area and service panel. Connections MUST be 4-wire: using /2+ground Romex to power ovens has always been illegal.
Hardwired large appliances cannot be on the above circuits, and need their own circuit(s) as needed based on their ampacity (x 1.25 if the load might conceivably run 3 hours or more, so not microwaves nor disposals). They can share circuits, but that "paints you into a corner" in the future, e.g. not allowing you to use a more powerful disposal because then disposal+dishwasher would exceed 20A. Generally best to have a dedicated circuit for each.
Only recommended: Dedicated circuit for refrigerator, set up with a single socket (not the normal dual socket) and labeled "Refrigerator only". No GFCI on this circuit!
Recommended: More than two Kitchen countertop receptacle circuits. One per countertop receptacle is ideal. That is because 2 cooking appliances (typically 1500W each) are typically too much for 1 circuit (2400W). This makes it easy for the chef to arrange appliances so they don't trip breakers.
The kitchen counter tops require 2ea 20 amp circuits the circuits have to be GFCI protected I prefer using normal breakers and putting A GFCI receptacle in the first position for each then if there is a fault you may hear it and can reset it without going to the service panel (if GFCI breakers are used the reset will be at that location). Additional devices may require more branch circuits but 2 circuits is the minimum. Dishwasher, compactors, built in microwave all have additional requirements but you did not specify anything other than kitchen. If my customers want to go cheap I put the kitchen refrigerator on one of the small appliance branch circuits and it is not GFCI protected but all the others are (receptacle protection not breaker in this case). Check your state code as individual jurisdictions have different rules (my state doesn’t require GFCI for built in dishwasher but the national code requires GFCI) knowing the rules for your area can save a lot of problems.
You need at least two 20A breakers/circuits for the countertops as well as dedicated for dishwasher and food disposal (depending on you local code and if load doesn’t exceed 80% of the total circuit capacity, those can be on the same 20A breaker). They also must be GFCI protected circuits. You can ether use 20A GFCI breaker or GFCI receptacle. Here is a good read for you